Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{-6z^3 - 18z^2 + 324z}{-7z^3 + 35z^2 + 42z}$
First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {-6z(z^2 + 3z - 54)} {-7z(z^2 - 5z - 6)} $ $ a = \dfrac{6z}{7z} \cdot \dfrac{z^2 + 3z - 54}{z^2 - 5z - 6} $ Simplify: $ a = \dfrac{6}{7} \cdot \dfrac{z^2 + 3z - 54}{z^2 - 5z - 6}$ Since we are dividing by $z$ , we must remember that $z \neq 0$ Next factor the numerator and denominator. $ a = \dfrac{6}{7} \cdot \dfrac{(z - 6)(z + 9)}{(z - 6)(z + 1)}$ Assuming $z \neq 6$ , we can cancel the $z - 6$ $ a = \dfrac{6}{7} \cdot \dfrac{z + 9}{z + 1}$ Therefore: $ a = \dfrac{ 6(z + 9)}{ 7(z + 1)}$, $z \neq 6$, $z \neq 0$